Assumption (WUU) implies that \(C(X^s_{\min },U^{[1]},d^{[1]}_{\min }) \ne \emptyset \)  and this is disjoint from the centre of the blow-up, since only points with \(U^{[1]}\) -stabiliser dimension equal to \(d_{\max }^{[1]}\)  are blown up. This proves (a).
To prove (b), first we claim that \(Z^{[1]}_{\min }\)  is not contained in the centre of the blow-up. Recall that the centre of the blow up is
\(D^{[1]}(X):=C(X^{[1],ss}_{\min }(X),\widehat{U}^{[1]}) = U^{[1]}C(Z^{[1],ss}_{\min }(X),U^{[1]}).\) 
By Lemma REF  (see also [1]} Lemma 5.2), we have for \(u\in U^{[1]}\)  and \(z\in Z^{[1]}_{\min }\)  that \(uz \in Z^{[1]}_{\min }\)  if and only if \(uz =z\) . Hence, we deduce that
\(D^{[1]}(X)\cap Z^{[1]}_{\min } = C(Z^{[1],ss}_{\min }(X),U^{[1]}).\) 
Thus it suffices to show that \(\hspace{0.83328pt}\overline{\hspace{-0.83328pt}C(Z^{[1],ss}_{\min }(X),U^{[1]})\hspace{-0.83328pt}}\hspace{0.83328pt}\subsetneq Z^{[1]}_{\min }\)  is a proper subscheme. But this consists only of points with \(U^{[1]}\) -stabiliser dimension equal to \(d_{\max }^{[1]}\) , by semicontinuity of stabiliser dimension. On the other hand, by (WUU) we know that \(C(Z_{\min },U^{[1]},d^{[1]}_{\min }) \ne \emptyset \) . Since \(Z_{\min } \subset Z_{\min }^{[1]}\)  by (QiS) (REF ), we see that \(Z_{\min }\)  is not contained in the centre of the blow-up and \(C(Z^{[1]}_{\min },U^{[1]},d^{[1]}_{\min }) \ne \emptyset \) . It then follows that \(Z^{[1]}_{\min }\)  is not contained in the centre of the blow-up, which proves (b).
To prove (c), the argument is the same as (b). By (WUU) we know that \(Z_{\min }\)  is not contained in the centre of the blow-up and by (QiS) (REF ) we have \(Z_{\min }\subset Z^{[i]}_{\min }\)  and \(Z_{\min }\subset Z^{(i)}_{\min }\) , which shows both \( Z^{[i]}_{\min }\)  and \(Z^{(i)}_{\min }\)  are not contained in the centre of the blow-up and proves (c).
For (d), as taking proper transforms preserves containment, Assumptions (REF ) and (REF ) of (QiS) for \(\widetilde{X}_{(1)}\)  follow from the corresponding properties for \(X\)  as (c) holds. For the non-degeneracy assumption (REF ) of (QiS), take any \(x \in X^{P-qs}\) ; this set is non-empty by (WUU). By (a), \(X^{P-qs}\)  is disjoint from the centre of the blow-up, so we can use \(\pi _1^{-1}\)  to lift \(x \in X^{P-qs}\)  to a unique \(\tilde{x} \in \widetilde{X}_{(1)}.\)  We claim that \(\tilde{x} \in (\widetilde{X}_{(1)})^s_{\min }\) . Let us use \(\tilde{p}\) , \(\tilde{p}_{[i]}\)  to denote the corresponding maps for \(\widetilde{X}_{(1)}\)  rather than \(X\) . To prove our claim first note that, since \(p(x)\)  is not blown up, we have \(\tilde{p}(\tilde{x}) \in Z_{\min }(\widetilde{X}_{(1)})\) . Furthermore, the linearisation on \(\widehat{X}_{(1)}\)  is an arbitrarily small perturbation of the one on \(X\) , so \(\tilde{p}(\tilde{x})\)  is \(R\) -stable, meaning that \(\tilde{x} \in (\widetilde{X}_{(1)})^s_{\min }\) . It follows that \(\tilde{x}\)  is a point witnessing non-degeneracy for \(\widetilde{X}_{(1)}\) , since otherwise \(\pi _1(\tilde{x})\)  could not lie in \(X^{P-qs}\) . Hence (QiS) holds on \(\widetilde{X}_{(1)}\) .
For (WUU), observe that \(\tilde{p}(\tilde{x})\)  has the same stabilisers as its image under \(\pi _1\) , since that image is not blown up. This shows that REF  still holds. Moreover, the same is true for each \(\tilde{p}_{[i]}(\tilde{x})\)  and its image under \(\pi _1\) , which together with the above argument gives \(\tilde{x}\in (\widetilde{X}_{(1)})^{P-qs}\) , so this set is non-empty. This proves (WUU) for \(\widetilde{X}_{(1)}\) , which completes the proof of (d).
The proof of statement (e) is the same as that of Lemma REF .
